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Note: For another, yet similar explanation of this, see http://mathmojo.com/interestinglessons/exponentsntothe0power/exponentsntothe0power.html
This was the question:
How do you get the answer 1 when you do x (any number) to the 0 power?
Professor Homunculus' answer:
(There is a second part to this answer at the end.)
In modern mathematics, we usually
use a base system to represent our numbers.
You know that we have units, tens, hundreds, etc. in out base-ten system.
Well, we also represent those columns in terms of 10 to the nth power. In other
words, the thousands column is represented by 10^3. So 8,000, for example is
8 X 10^3.
8,300 would be (8 X
10^3) + (3 X 10^2).
8,320 would be (8 X 10^3) + (3 X 10^2)
+ (2 X 10^1).
Now comes the problem. You
see how the base stays the same, and the exponent gets smaller? To represent
the units column, mathematicians have accepted the convention that 10^0 will
always equal 1.
That keeps the rule going. So 8,325 would be (8 X 10^3) + (3
X 10^2) + (2 X 10^1) + (5 X 10^0).
Because other bases systems, (base 2, base 3, etc.) work the same way, we have
further accepted the convention to be n^0 always equals 1, no matter what the
base.
Take the binary (base 2) number 1011 for example. What that means is
(1 X 2^3) + (0 X 2^2) + (1 X 2^21) + (1
X 2^0)
That is the same as 8 + 0 + 2 + 1
which is the number 11 in our normal base 10 system.
As long as we keep n^0 = 1, then the units column of any base will always mean
how many ones there are in it.
Math is a network of "conventions" mankind has accepted to make it
work. It is based on rules and axioms that are the most "convenient".
They may seem hard to figure out at first, but when you get down to it, the
rules that we use are basically the best we can come up with to get the things
done that we want to accomplish.
Every once in awhile some genius comes up with a rule that makes something even
simpler than how we have been doing it up until now, and that becomes the new
convention. But that rule has to be solidly based on what has come before, and
may not break any of the other rules.
Maybe someday you will be one of
the people who comes up with something that explains, or enables something that
until now was done by a convention that needed improving.
Have fun!
Professor Homunculus
Someone replied:
I got a thoughtful question from a reader, which addressed the Idea of "conventions." This was the letter:
Exponent n^0 =1 is stated by 'Professor Homunculus' to be a "convention" to make the bases system work. However, some of us need more of an explanation than 'just because it makes it work'. I was wondering if you would be able to give me a more concrete response to explain this 'convention'.
Professor Homunculus replied:
My answer was more prolix about "conventions." It was
relevant, but I did miss a good point, which I had also missed in the original
explanation.
Your question is great. The answer may be surprising, though.
Here's my take on it, but please remember that I am not "the authority."
I am not even an "official mathematician," I can just calculate in
my head faster then most of them. My interest is in basic math, arithmetic,
and mathematical philosophy.
So my answer may not be "cut and dry." But it will make sense in
a more meaningful way, if you digest it and mull it over for a while.
Actually, all the rules and "facts" we use in mathematics are based
on some fundamental assumptions, called axioms (or axia). We assume that the
whole numbers go on infinitely. We can't prove it empirically, because we don't
have the time. But we do know that for any number I say, you can say a higher
one. So we "assume" (using a kind of logic called "induction")
that the numbers never end.
Now, as I see it, there really are no such things as numbers. I mean, nobody
ever saw a 4 walk down the street. (I have seen a 10 once, and I married her,
but that is something else.) The number 4 is just a symbol for an Idea. We
could have called that symbol "snsasyer" or "ignatz," but we chose
to call it "four."
To a German it is "vier", and "four" has no meaning. Each
are conventions we use to make things work. If we didn't name them something,
it would be harder to use them. We'd have to hold up four fingers and say, "This
many."
Holding up four fingers and saying "this many" still works, though,
although it is not as convenient as using the number 4. That is what it is all
about - convenience. There are some rules about making things convenient, though.
First is that one of the aims should be to keep things consistent. For example,
if we normally count by tens, as "ten, twenty, thirty," etc, but all
of a sudden we say "fourscore" instead of eighty, it still works,
but it is not consistent. So we "conventionally" say "eighty."
(Did you know that in some languages, like French, they still say "fourscore"?
They call it "quatre-vingt" - literally "four twenty." It
confuses the heck out of beginning French students, simply because it is unconventional.
OK, so far, so good, I hope. One more, VERY important rule about convenience,
is that no convention can contradict any existing conventions. That would mess
up the entire fundament of math.
Here is an example. If you try to divide a whole number by 0, you might expect
to get something meaningful, like 0, or the original number. You don't, though.
Why?
Because one accepted convention in arithmetic is that if you divide
some dividend by a divisor, you will get a product, and if you multiply that
product by the original divisor you will get the original dividend. Or, if you
multiply the product by the original dividend you will get the original divisor.
I'm sure you know this rule.
So, according to that rule, if 8/0=0, then 0*0 would have to equal 8, and that
is clearly not the case. So we have the convention that "you cannot divide
by zero." The truth is you can divide by zero, but any answer would be
nonsense. So what do we do with this dilemma? We say, "Anything divided
by 0 is "undefined."" That is also a convention. It is the only
way (or at least the best way) to solve the dilemma, without offending any other
conventions.
Mathematicians make up conventions all the time. As long as they keep things
non-contradictory of other conventions, it is legitimate. Some conventions take
hold, some don't. Some are more useful than others. Some become useful many
years after they are invented. For example, the western world didn't even have
the convention of using 0 as a number for a very long time. It was invented
somewhere in India, over a thousand years ago. Many westerners thought it was
strange to have a symbol for nothing, and they did not adopt it, (to their detriment)
until about the 15th century.
So math is a fluid, ever changing challenge. It is not a bunch of laws which
are "true" and came down with Moses from the mountain, or something
like that. It can change any time someone finds something more convenient than
the way things are done now, and find an application for it.
They laughed at George Boole when he developed Boolean logic. No one found a
significant use for it during his lifetime. It "didn't make sense,"
was "useless" etc.
Now, the entire field of information and computer science would still be in
the Kindergarten stages without it.
The point I am getting at is that conventions are about as concrete as things
get (besides axioms) in math. To expect more is to be stuck in the intuitive
(but not very helpful) mode of expecting math to be "concrete."
If I may make one analogy before I bore you to death, here it is:
Imagine a discussion in which some child says, "Daddy, were we right or
wrong to fight in the Vietnam war as long as we did?" Now I have an opinion,
and you have an opinion, probably. But neither is probably "the truth."
Only in juveile fiction like "The X Files" is it that "the truth
is out there." The truth is a subtle and nuanced concept, and the truer
it is, the less concrete it tends to be.
That is just the humble opinion of a "street-philosopher/mathematician,"
but I think you will find a lot of mathematicians and philosophers who see it
similarly. Unfortunately, most of them don't teach in the public school system.
It is easy to imagine that this will be hard to explain to children or adolescents.
It usually is. Good luck!
I hope this helped.
Brian (a.k.a. Professor Homunculus)
Thank you for your answer!
It was also explained to me that 7 squared divided by 7 squared would be the same as 7 (2-2) over 7 (2-2), making it the same as 7 to the 0 power, which would equal one, which in turn made some sense to me.
I am not very good when it comes to math and usually need as much explanation as I can get!
Professor Homunculus replied:
You are so right. I can't believe that I neglected the basic math to the answer, instead of giving you the history of life, the universe and everything!
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"Even when all the experts agree, they may well be mistaken."
- Bertrand Russell
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Another note: There is an interesting post concerning this
and similar issues at: http://www.mathmojo.com/chronicles/the-power-of-zero
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